Moduli of relatively nilpotent extensions:
RIMS Publications, Paper on three lectures at RIMS, October 2001

p-Frattini-cov.html is a short exposition on the motivation for the universal p-Frattini cover of a finite group G. We suggest visiting that first.

§2 of this paper includes the explicit (and general) construction of the first Frattini module that defines the universal p-Frattini cover of G. §I outlines the most precise available description of the p-Frattini module for any p-perfect finite group G=G₀ (Thm. 2.8), and therefore of the groups G_k,ab, k ≥ 0, from which we form the abelianized M(odular) T(ower). §II points to the difficulties in being explicit, and to examples where it has been done.

§4 of the paper includes a classification of Schur multiplier – Z/p – quotients. §III reviews that. One type from that classification stands out: Schur quotients that are antecedent. The special case of Spin covers of the alternating group, shows how one (tiny; Z/2) cover of an alternating group plays a mighty role in the history of mathematics. Further, as an antecedent, it keeps reappearing in any tower once it appears in a low level.

§IV reviews later developments, from which we figure two points related to Braid orbits:

Whether there is a non-empty MT over a given Hurwitz space component at level 0; and
whether all cusps above a given level 0 o-p' cusp are p-cusps.

The diophantine discussions of §5 remind how Demjanenko-Manin worked on modular curve towers, to contrast why we still need Falting's Thm. to conclude the Main MT conjecture when the p-Frattini module has dimension exceeding 1 (G₀ is not p-super singular).

§V enhances this with comments on the motivic pieces of abelian varieties that appear in MTs. It appears from this that unless those motivic pieces have dimension 1, a Demjanenko-Manin approach won't work to conclude rational points disappear at high levels.

I. The characteristic p-Frattini module, M₀=M_G_,0 of G, a group of order divisible by a prime p:

Background for the statements made here can be found in [FrJ, Chap. 21] on the universal Frattini cover of any profinite group. The Frattini subgroup, Φ(G) of G has an abstract definition as the intersection of all maximal proper subgroups. For, however, a (pro-)p group P, it is the (closed) subgroup generated by commutators, aba^-1b^-1, and pth powers a^p, a, b ∈P.

I.1. The p-group case: Let P be a p-Sylow of G, and M_P,0 is the characteristic p-Frattini module of P. A brief explanation of a section in the paper explains M_P,0 this way. Let P^* be a pro-free group of the same rank, rk_P as P. Denote by ψ_P: P^* → P a corresponding surjective homomorphism.

Then, the kernel, ker(ψ_P), of ψ_P is also pro-free. Its rank, by the Schreier Thm., is rk_P,0=|P|(rk_P – 1)+ 1. This is also the rank of M_P,0=ker(ψ_P)/Φ(ker(ψ_P)), which is a Z/p[P] module. Then, G₁(P)=P^*/Φ(ker(ψ_P)) is the universal exponent p extension of P. It has the natural cover ψ_P,0: G₁(P) → P.

I.2. The general finite group case: Let N_P be the normalizer in G of P. Then, consider the module induced from M_P,0 in going from N_Pto G, Ind_N_{_P}^G (M_P,0). Thm. 2.8 identifies M_G,0 as the summand whose restriction to P contains M_P,0 as a summand.

Recall the main properties of M_G,0:

It is an indecomposable Z/p[G] module: has no nontrivial Z/p[G] module summand.
It is the kernel of a covering group ψ_G,0:G₁ → G versal for all covers H → G with kernel a Z/p[G] module.
The lift of any element of order p in G to G₁ has order p².

Given a clear understanding of P, this construction gives a handle on M_G,0, reasonably putting bounds on its rank.

I.3. Refining the construction of M_G,0: For each p-Sylow P, we have defined G₁(P) → P ≤ G. Further, we can pull P back in G₁(G)=G₁, to get ψ_G,0^-1(P) covering P. Applying the universal extension property of §I.2 then induces an injection α: M_P,0 → M_G,0.

More generally, any conjugation by g ∈G will map P to gPg^-1, and induce another map α_g: M_P,0 → M_G,0. (Similarly, by an outer automorphism of G.) In the construction of §I.2, it often occurs that a subgroup of G, N_P^h, properly larger than N_P, acts to preserve M_P,0. §II gives examples where the next lemma applies to precisely identify M_G,0.

Lemma: The images of {α_g}_{g ∈G} generate M_G,0. Define N_P^h to be those g ∈G with the same image as α. If we replace N_Pin Ind_N_{_P}^G (M_P,0) by N_P^h, the induced module gives precisely the module M_P,0.

II. Examples of p-Frattini modules:

III. Z/p quotients of Schur multipliers and their antecedents:

IV. Components and cusps:

V. Why there is yet no replacement for using Falting's Thm. in the Main Modular Tower Conjecture: